Significance of $i$ in the Schrödinger equation There's an imaginary $i$ in the Schrödinger equation, which I guess is to define the position of the particle in a space-time involving a complex function. But what is the real physical significance of $i$ in the equation?
 A: The Schrödinger equation is related to the Schrödinger-Pauli equation (which has spin).
And there are relativistic equations too. And they all have algebraic objects that add and multiply like $\pm i$ do. So you can look at those equations and their factors and terms and see what they represent and look at the nonrelativistic and constant spin limits.
One geometric description of the relativistic wave equation for an object of spin 1/2 is that it takes a reference spatial plane (which adds and multiplies like $\pm i$) and gives it a phase rotation in its plane, scales it by a positive scalar, rotates it into an arbitrary attitude plane (possibly different than the original reference plane's attitude), and gives it a relativistic boost to put the (now rotated) reference spin plane into an arbitrary plane of simultaneity. That covers all the degrees of freedom of the equation and does it will things that algebraically act like those parts of the equation and geometrically do those things to geometric objects in a spacetime.
Then the non relativistic limit has a constant spin limit that is the Schrödinger equation where the $i$ survives in the equation as the initial arbitrary reference plane.
That doesn't mean that's what the $i$ is. But for some problems you are using the Schrödinger equation for a spin 1/2 particle in a a nonrelativistic situation with a constant spin. In which case it is harder to think it can represent anything else.
